yiton t. yan- collective acceleration of protons by the plasma waves in a counterstreaming electron...

8/3/2019 Yiton T. Yan- Collective Acceleration of Protons by the Plasma Waves in a Counterstreaming Electron Beam

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COLLECTIVE ACCELERATION OF PROTONS BY THE PLASMA WAVES

IN A COUNTERSTREAMING ELECTRON BEAM

Yiton T. Yan

Applied Theoretical Physics Division, Los Alamos National Laboratory, Los Alamos, N. M. 87545

Abstract

A novel advanced accelerator is proposed. Thecounterstreaming electron beam accelerator relies onthe same physical mechanism as that of the plasmaaccelerator but replaces the stationary plasma in theplasma accelerator with a magnetized relativisticelectron beam, drifting antiparallel to the drivingsource and the driven particles, as the wave

supporting medium. The plasma wave in a counter-streaming electron beam can be excited either by adensity-ramped driving electron beam or by properlybeating two laser beams. The fundamental advantagesof the counterstreaming electron beam accelerator overthe plasma accelerator are a longer and tunable plasmawavelength, a longer pump depletion length or a largertransformer ratio, and easier pulse shaping for thedriving source and the driven beam. Therefore, the

energy gain of the driven particles can be greatly

enhanced whereas the trapping threshold can bedramatically reduced, thus admit ting the possibilityfor proton acceleration,

Introduction

Using plasma waves to accelerate electrons haveattracted extensive attention ever since the first

proposal of the plasma beat-wave accelerator (PBW A) by

Tajima and dawson in 1979.l -By properly tuning twolasers they demonstra ted that a plasma wave with phasevelocity near the speed of light c can be generated inthe plasma. A bunch of electrons can then ride thewave electric field and accelerate to high energy.Another promis ing scheme that can also generaterelativistic plasma waves, the plasma wake-field

accelerator (PWFA), has also been proposed ;ecently.2

The PWFA differs from the PBWA in that it replaces thetwo laser beams in the PBWA with a relativisticelectron beam as the driving source. The basic

advantage of using the plasma waves to accelerate

charged particles is that there is no electrical

breakdown since the plasma is already fully ionized.Thus the acceleration gradient is only limited by theplasma wave breaking field which can be an order of aGev/m at present.3

Although both the PBWA and the PW FA are veryattractive, there are some fundamen tal shortcom ingsassociated with them, namely the inherent short plasmawavelength, the inherent short pump depletion length,and the difficulty for pulse shaping of the drivingsource and the driven beam. Aiming at improving thesedrawbacks in the plasma accelerators, the author

proposes using a counterstreaming electron beam as therelativistic plasma wave supporting med ium for chargedparticle acceleration.

Similar to the plasma wake-field accelerator, the

plasma wave in a counterstreaming electron beam can beexcited by a shaped driving electron beam. This is

called the counterstreaming electron beam wake-field

accelerator (CWFA ). Similar to the plasma beat-waveaccelerator. the plasma wave in a counterstream ingelectron beam can also be excited by properly beatingtwo laser beams. This is called the counterstreamingelectron beam beat-wave accelerator (CBWA). Excitingthe plasma wave ina counterstreaming electron beamhas several fundamen tal advantages over the plasmaaccelerators: (1) a longer plasma wavelength, (2) alonger pump depletion length or a larger transformerratio, and (3) easier pulse shap ing. With a longer

plasma wavelength, beam loading onto the plasma wavecan be made easier; phase slippage between the driven

particles and the plasma wave is reduced; energyspread among the accelerated particles is madesmaller; and the particle trapping threshold is lower.With a longer pump depletion length or a largertransformer ratio, the driven beam can be acceleratedfor a longer distance in a single acceleration stage;thus a larger energy gain per acceleration stage canbe achieved. Easier pulse shaping particularly meansthat the sharp tail cutoff of the driving source canbe made easier because of the Lorentz contractioneffects. The most interesting characteristic of thecounterstreaming electron beam accelerators is thatthey (both the CWFA and the CBWA) admit thepossibility of trapping and accelerating moderatelyrelativistic protons, a step impracticab le with theplasma accelerators.

Basic Feature of the Plasma Waves

Consider a counterstreaming electron beam of restframe density no and relativistic velocity vd' -pdCe,

as the plasma wave supporting medium. For-the CWFA,let us assume that the driving source is an ideal"doorstep" linearly risisng density and sharp tailcutoff electron beam of length Lb, peakndensity nb,and with a relativistic velocity Vb"pbce, - as thathas been considered in the -plasma wake-fieldaccelerator.4,5 For the CBWA, let us assume that thedriving source is a pulse of light consisting of twolaser beams with respective frequencies ~1, w2 andfields El, E2 moving in the x-direction and beating atthe frequency Au=01 - w2 = up/f. Here op=(4xnoe2/m)1/2is the proper (rest frame) p lasma frequency of thecounterstreaming electron beam; m is the rest mass of

an electron; and f is denfined as the counterstreamingfactor and is given by

f = -,d (1 + &Bd) - z-i'd, (in the CWFA)or

f = Yd (1 + Pd) - ?Yd, (in the CBWA)

Oni? way to examine the counterstreaming electronbeam accelerator is to analyze this accelerator inthe rest frame of the counterstreaming electron beamsince, in this frame, the basic physical mechanism

of the PWFA or the PB WA is recovered so one candirectly apply those results obtained for the plasma

wake-field accelerator, 4.5 or for the plasma beat-waveaccelerator.1~6~7 Lorentz transforming those resultsobtained in the counterstreaming electron frame to the

laboratory frame, one then obtains the following:8,9

Phase velocity: The phase velocity of the plasmawave in the counterstreaming electron beam is given by

lip = vb, (in the CWFA)or (1)

VP = (1 "p2/w1+2c, (in the CBWA)

where 01 > ~2 2> we has been implicitly assumed

Amplitude: The electrostatic amplitude of theplasma wave in the counterstreaming electron beam isgiven by

E, = cE*, (2)

175 CH23R7-9/87/0(XH)-O17.5 p;, .W ~9 IEEE

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PAC 1987



where Em = mcwp/e is the plasma wavebreaking field inthe coL:nterstreaming electron beam; and c is given by

E = frlb&, (in the CWFA)"II

E = (y a1a2)1/3,(3)

(in the CBWA)

where c < 1, that is, fnb<no, or ala2 < 3/16, - hasbeen implicitly assumed; and al,2 = eEl,2/mcwl,2 is ameasure of the laser pump intensity.

Wavelength: The plasma wavelength in the counter-streaming electron beam is given by (for both of theCWFA and the CBWA)

lP = fx; = f(2rrc/wp). (4)

where X'!

is the plasma wavelength referred to the restframe 0 the counterstreaming electron beam.

Since the acceleration gradient is basicallylimited by the plasma wave-breaking field Em, for

purposes of comparison it is reasonable to assume thatE is a chosen constant. Therefore, for a given rest

fraclr density no of the counterstreaming electronll<'RT , from Eqs. (l)-(4), one finds that the plasma

wavelength is proportional to f whereas the amplitudecllld the ?hZiSe velocity of the plasma wave n r 2

indrpendent of f. Note that f = 1 (,!?d = 0) stands for

the plasma accelerators.

Pump Depletion

In general, any accelerator suffers pump depletionas the accelerating field 10.Ses energy to the

accelerated particles. This is usually referred to asthe beam loading energy loss. Using plasma wave toaccelerate particles, however, there is another

significant channel of energy loss, that is, the

plasxa waves extract energy from the driving source SC

that r?ventunlly the energy conta ined in the drivingsource will be depleted. The pump depletion length isthus defined as the length the driving source travelslefore its energy gets depleted. this is the maximum<istance the driven particles can be possibly

sccelerated in a single acceleration stage.From energy conservation, thelength can be roughly given by8,g

Pump depletion

3r Lp = R(x,-l)mc2/e+, (in the CWFA)

(5)Lp = 4f(C/Wp)(~12/Wp2)(ala2/4)-1/3, (in the CBWA)

@here R is the transformer ratio in the CWFA and is

given byRa10

R=E+/E.= 2TtLbjXp=0.03f~yb/E, with I < 1. (6)

Here E+ ar.d Em are respectively the maximalaccelerating electric field behind the driving

electron beam and the maximal decelerating electric

field within the driving electron beam. Keglecting

phase slippage effects, the transformer ratio actuallyrepresents the ratio of the maximum energy gain AW ofa R accelerated particle with cha rge Iq/=e to theinitial energy of a driving electron, (yb-l)mc*. Notetha: Q is the length control factor of the drivingelectron beam. The tracsformer ratio R is limited(Q < 1) because the driving electron beam length cannot be to long; otherwise the two-stream instabilitywill be strong enough to degrade the driving electronbeam. The reason why the transformer ratio R isproportional to f is because the Lorentz effects

resulting from the relativistic streaming of thecounterstrenrr ing electron beam can help suppress theharmflu two-stream instability.8 Similarly, the reasonwhy the pump depletion length in the CBWA is

proportional to f is because the Lorentz contraction

effects can accumula te more laser energy which is

spreaded in a longer distance in the laboratory frameinto a useful si;orter distance in the counterstreamingelectron frame.

Energy Gain

In the CWFA, since the pump depletion length givenby Eq. (5) is not long enough to have appreciablephase slippage effects, the energy gain of a drivenparticle with charge q and mass m

approximately by8

Pcan be given

AW = qE+$ = 0.03f@(q/e)~b*mC*/c. (7)

On the other hand, in the CBWA the pump depletionlength given by Eq. (5) is long enough that the energygain is basically limited by the phase slippage and isgiven approximately by8

AW = 2r,@,(l - (-2) :/2m c2. (8a)

Here E = 1 t rf(q/e)(m/mp)EBprp, with 0 < r < 2,depending on the loading phase and the allowable phaseslippage (normally, I- = 1 because the allowable phase

slippage is usually one fourth of the pL35ITEl

r~a~~~g~~:at~is;i‘bP'f",;,t~~ ;I!= (1 - p;)-liz = wl/wpthe plasma wave. For

E >> 1 (always true for electron acceleration), Eq.(8a) can be simplified as (r = 1, q/e=l)

AW = 2fy$zmc'. (8b)

Trapping Threshold

If a driven particle of charge q and mass mp, witha velocity lower than the phase velocity of the plasmawave at the beginning of acceleration, can catch upthe phase velocity of the plasma wave in a shortperiod during which the relativistic factor of the

plasma wave yp changes little, The trapping thresholdis given approximately by8

wth = ( YP( [ 1 - /?,(I- - [-')'/'] - 1) mpc2. (9)

This is generaliy true for electron acceleration inboth the CW FA and the CBWA but is only true in theCBW;: for proton acceleration provided that wl > w2 >z

wP. As for proton acceleration in the CWFA,- if theprotons are initially loaded onto the plasma wave inphase with the acceleration gradient near E, and theallowed phase slippage distance uXp for the plasmawave to outrun the protons is rruch smaller than xthe trapping threshold can be calculated to be8

P'

“th =/h ( l+;(l+& ))I'*-h-l 1

‘\ 2hM/Rm - 1 ~b-l(Mc', (10)

where h = ?afacyb(m/M) and M is the proton rest mass.

Proton Acceleration

Since the energy gain of the driven particles isbasically proportional to the counterstream ing factor

f as shown by Eqs. (7) and (8) and the trappingthreshold can be appreciably reduced with a large f ascan be numerically calculated from Eqs. (9) and (lo),8it would be interesting to consider the possibilityfor proton acceleration. Indeed. based on recent

advances in the electron beam technology, one might

consider a counterstreaming electron beam of energy5O?leV ( d = 100, f = 200) with rest frame density no =1012cm- 3 (Em = lOOMV/m). Taking CWFA for example, one

migh t also consider a shaped driving electron beam of

energy 5OMeV (yb = loo),= 2 I( 109cm-3

with peak density nb = n,c/fif one chooses E to be 0.4. The maximum

proton energy gain per acceleration stage can then be

176

PAC 1987



calculated from Eq. (7) to be ~w=42VGeV. Furthermore,if the length control factor of the driving electronbeam, 3, is chosen to be the maximum allowable value(111= 1) and a reasonable value of a = 0.18 is used,from Eq. (10) the proton trapping threshold can becalculated to be Wth = 6 GeV. The trapping efficiencywould then be

11 = AW/(hW + Wth) = 42GeV/(42GeV + 6 GeV) = 88%.

HOVEVE!r) this requires a long driving electron beamand a long counterstreaming electron beam (about 1000meters)8 that may not be achievable with currenttechnical ability. Therefore, for a practicalexperiment to be performed now, one may have to choosea small length control factor of the driving electronbeam, 1. For example, if Q=O.O5, one has Aii = 2.1GeV,wth = 2.4, and 17 = 47%, and that both the drivingelectron beam and the counterstreaming electron beamare within 100 meters. 8 One can even choose a lowervalue of I than 0.05 to further reduce the required

length of the driving electron beam and thecounterstreaming electron beam. Although this willfurther reduce the energy gain AW per accelerationstage, the required initial energy Wth is also reducedso that the trapping efficieny q is essentially notmuch affected. 8 On the other hand, let us consider thecase of using a 50MeV driving electron beam but

replacing the counterstreaming electron beam with astationary plasma (PWFA), that is f =l. Then even ifC-l, the energy gain is only A W = 0.21GeV whereas thetrapping threshold becomes Wth = 17GeV; thus thetrapping efficiency becomes extremely low (r,~ = 1.2%).These numbers clearly show why the counterstreamingelectron beam accelerator migh t be considered as acollective proton accelerator whereas the plasmaaccelerator is indeed impracticable for accelerating

protons.

Shortcomings

One shortcom ing of using a relativistic counter-streaming electron beam as the charged particleacceleration medium is that the kinetic energy of thecounterstreaming electron beam is not transferrable to

the accelerated particles. However, as has been donefor free-electron lasers, there may be ways ofretrieving this energy, or it migh t be used for otherstages of particle acceleration. This remains furtherinvestigation. Another shortcoming is the presentinability of generating a high-energy, high-density ,and long electron beam that can have a rest framedensity equal to that can be achieved for a plasma atrest, Thus the achievable acceleration gradient in thecounter-streaming electron beam accelerator is actuallylower than that can be attained using a plasmaaccelerator. Nonetheless the attainable accelerationgradient (Em = lOOMV/m) in a counterstreaming electron

beam is still comparable to or larger than those can

be obtained via conventional accelerator means.

Conclusions

In conclusi.on, the author has proposed a collectiveplasma WF3VZ2 accelertor using a counterstreamingelectron beam as the wave supporting medium. Theplasma wave in a counterstreaming electron beam caneither be excited by a shaped driving electron beam,just as the plasma wake-field accelerator does, or beexcited by beating two laser beams, just as the plasmabeat-wave accelerator does. Despite the above-mentioned shortcomings, the fundamental advantages ofthis newly proposed accelerator over the plaSmaaccelerators, - e.g. a longer plasma wavelength, alonger pump depletion length or a larger transformerratio, and easier pulse shaping, particularly theability for accelerating protons, make this an

interesting scheme worth pursuing further. Finally,The author would like to emphasize that the effectivesharpness of the driving electron beam tail cutoff(counted in the counterstreaming electron beam frame)in the CWFA is enhanced by a facotr of f2 because ofthe Lorentz constraction effects.8 This will greatlyimprove the performance of the wake-field accelerator.

Acknowledgements

The author would like to thank Prof. J. M. Dawson,

Dr. D. F. DuBois, Dr. D. Forslund, Dr. M. Jones, Dr.T. Katsouleas, Dr. R. Keinigs, and Dr. C.J. McKinstriefor helpful conversations. This work was performedrnder the auspices of the U.S. Department of Energy.

References

1. T. Tajima and .J. M. Dawson, Phys. Rev. Lett. 43,267 (1979).

2. P. Chen, J. M. Dawson, R. W. Huff and T.Katsouleas, Phys. Rev. Lett. 54 693 (1985).

3. C. E. Clayton, C. Joshi, C. Darrow, and D.

Umstadter, Phys. Rev. Lett. 54, 2343 (1985).4. P. Chen, J.J. Su, J.M. Dawson, K.L.F. Bane,

and P.B. Wilson, Phys. Rev. Lett. 56, 1252 (1986).

5. T. Katsouleas, Phys. Rev. e, 2056 (1966).6. M. N. Rosenbluth and C. S. Liu, Phys. Rev. Lett.

29, 701 (1972).7. W'. Horton and T. Tajima, Phys. Rev. A3$. 4110

(1986).8. Y. T.'Yan, Los Alamos National Laboratory Report

No. LAUR-87-922 (1987), to be published.9. Y. T. Yan, et al., to be published.10. Y. T. Yan, R. Keinigs and M. E. Jones, to be

published.

177

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yiton t. yan- collective acceleration of protons by the plasma waves in a counterstreaming electron...

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